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On Absolute Continuity of Feller's One-Dimensional Diffusion Processes

✍ Scribed by Jürgen Groh

Publisher
John Wiley and Sons
Year
1984
Tongue
English
Weight
500 KB
Volume
116
Category
Article
ISSN
0025-584X

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✦ Synopsis

A class of FELLER'S one-dimensional continuous strong MARKOV processes generated by the generalized second order differential operator D,,,DZ is considered. In the case of natural boundaries of the state space R and an identical road map s(z) = x, these diffusion processes are martingales. In a first part of this note some earlier results concerning the representation of such processes as weak solutions of stochastic differential equations are improved. The second part concerns with diffusions absolutely continuous with respect to a given one, determined by the generator DmDZ+. Such absolutely continuous diffusions on the line were first described analytically by S. OREY in terms of the corresponding speed measures and road maps. By the aid of the derived stochastic equation an explicit expression for the corresponding RADON-NIKODYM derivatives is possible. This allows a characterization of diffusions with non-identical scale functions by stochast,ic differential equations.

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